International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019

On Prime Numbers and Related Applications

V Yegnanarayanan, Veena Narayanan, R Srikanth

 Abstract: In this paper we probed some interesting aspects of Our motivating factor for this work stems from the classic primorial and factorial primes. We did some numerical analysis result of called Prime Number Theorem which beautifully about the distribution of prime numbers and tabulated our models the statistical behavioral pattern of huge primes viz., findings. Also, we pointed out certain interesting facts about the utility value of the study of prime numbers and their distributions chance for an arbitrarily picked 푛휖푁 to be a prime number is in control engineering and Brain networks. inverse in proportion to log(푛). Keywords: Factorial primes, Primorial primes, Prime numbers. II. RESULTS AND DISCUSSIONS I. INTRODUCTION A. Discussion on Primorial and Factorial Primes The irregular distribution of prime numbers makes it Our journey is driven by fascinating characteristics of difficult for us to presume and discover. Euclid about 2300 primordial primes and factorial primes. A primordial years before established that primes are infinitely many. Till function denoted 푃# is the product of all primes till P and date we are not aware of a formula for finding the nth 푛 includes P. The factorial function is 푛! = 푖=1 푖. A natural prime. It appears as if they are randomly embedded in N and curiosity pushes us to probe large primes. In order to do this, may be because of this that sequences of primes do not tend we need to get an idea by analyzing numbers such as to emanate from naturally happening processes. We 푛! + 1, 푛! − 1, 푃# + 1, 푃# − 1 . As these numbers have large constantly use computer encryption whenever we factors it is quite evident that their density of primes is more communicate our credit card information to Amazon or than the average. The following facts are well known. One logging into our bank account or posting an encrypted email. can also see [1]-[4] for more. It simply implies that we are using prime numbers and rely on 1 their properties for protection of life in this cyber-age. Carl (a) The density of primes close to M is log 푀. Sagan in his novel “Contact” remarked that prime numbers 푁 푁 are cool. He further said in that book that aliens prefer to (b) 푁! ≈ encode their message as long string of prime numbers. Many 푒 (c) If 푝휖 푁, 푁! + 1 , 푝 prime and 푁! + 1 ≢ of the cryptographic algorithms depend on finding large 0 (푚표푑 푝), then 푁! + 1 is a prime. prime numbers. So, if one did not perform the primality testing with a proper prime gap, then the algorithm itself The task of determination of huge primes comprises two would consume a very long time to run. Thus, one of the distinct steps. The first step is to determine the chance of interesting questions arising in the study of prime numbers is being a prime and the second step is testing for the primality. the distance between two successive primes. According to A number A is declared to have more chance of being a prime Grandville, understanding both the small and large prime if (푐퐴−1 − 1) ≡ 0(푚표푑 퐴). Here the option for c is not that gaps is extremely difficult and hence would not be answered pertinent except in typical situations where c is restricted until another era. from assuming certain forbidden values. It has already been Lately problems in number theory are handled with the estimated that the mathematical expectation of the number of math software tools such as maple or MATLAB. These tools 퐾2 2푑퐾 퐾2 primes from 퐾1 푡표 퐾2 is 퐼 = = 2 log( ) and hence processes complex computations with ease reducing 퐾1 퐾 퐾1 it takes significantly longer time to determine the succeeding voluminous labor. It is incredible to note that Goldbach’s factorial prime. Next, about the testing of primality, if a huge conjecture was found to be true through numerical techniques 5 for every number that is congruent to zero (mode 2) but less integer with more than 10 digits has higher chance of being a prime then in reality it mostly turns out to be so. A nice than or equal to 4 × 1018 . reference for practical tests for primality of huge numbers 5 with 10 or above digits is [5]. Starting with 2, as first prime number, it is customary to let 휋(푥) as number of prime less than or equal to x; 휗(푥) as the logarithm of product of all primes less than or equal to x, 훹(푥) as the logarithm of the lcm of all positive integers less than or equal to x; Also we let Revised Manuscript Received on October 05, 2019. 휋 푥 = 휗 푥 = 훹 푥 = 0 푖푓 푥 < 2 푎푛푑 푥휖푅 . In [6] the Veena Narayanan, Department of Mathematics, SASTRA Deemed authors have obtained nice upper and lower bounds for University, Thanjavur, India. Email: [email protected] V Yegnanarayanan, Department of Mathematics, SASTRA Deemed 휋 푥 , 휗 푥 푎푛푑 훹 푥 . University, Thanjavur, India. Email: [email protected] R Srikanth, TATA Realty-SASTRA Srinivasa Ramanujan Research chair for Number theory, SASTRA Deemed University, Thanjavur, India. Email: [email protected]

Published By: Retrieval Number: L36521081219/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.L3652.1081219 4150 & Sciences Publication

On Prime Numbers and Related Applications

A natural approach to determine an approximation for the C. Numerical Analysis on Distribution on Primes 푥 푓(푦)푑푦 Here we provide a heuristic estimate for the primes of the summation 푓(푝) p ≤ x is to let it as equal to 2 . In log 푦 푘 푥 푑푦 푥 form 푦 = 2 푥 + 1; 푥 ∈ 푃 − 2 ; 푥 ′푠 are distinct. If view of this, 휋 푥 푎푛푑 휗(푥) becomes 푎푛푑 푑푦 푖=1 푖 푖 푖 2 log 푦 2 π(n) = #{k ≤ n : k is prime} defines the number of primes not which is close to x. It has been universally agreed that the exceeding a fixed real number n, then the Prime Number 푛 following numerical approximations or estimations are Theorem (PNT) states that π n ~ where ln is the natural 푥 ln 푛 accurate and valid for almost all values. That is, log 푥 1 + logarithm. The Table I interprets this. 풏 12log푥<휋푥<푥log푥(1+32log푥) where 푥≥59 for lower Table-I: Calculation of 흅(풏)~ 푥 퐥퐧 풏 inequality and 푥 > 1 for upper inequality. 1 < 풏 log 푥 − n 흅(풏) 2 퐥퐧 풏 푥 휋 푥 < 3 for 푥 ≥ 67 for the lower inequality 10 4 4.3429 log 푥 − 2 50 15 12.7811 1.5 푥 and 푥 > 푒 = 4.48 for the upper inequality. log 푥 < 100 25 21.7142 500 95 80.4559 휋 푥 < 1.25506 푥 where 푥 ≥ 17 for lower inequality log 푥 1000 168 144.7648 1 and 푥 > 1 for the upper inequality. 푥 1 − 2 log 푥 < 5000 669 587.0478 휗 푥 < 푥 1 + 1 where 푥 ≥ 563 for the lower 2 log 푥 10000 1229 1085.7362 inequality and 푥 > 1 for the upper inequality. 1 −

푙표푔푥푒−푙표푔푥푅푥<휗(푥 ≤훹(푥) and 50000 5133 4621.16678 푙표푔 푥 − 휗 푥 ≤ 훹 푥 < 1 + 푙표푔 푥 푒 푅 푥 where 푅 = 100000 9592 8685.88964 515 Now define π*(n) = #{y ≤ n: y is prime} where y = 2x1x2 · ≈ 217.51631 and 푥 ≥ 2 for lower ( 546 − 322) · · xk + 1; xi ∈ P − {2} are distinct. Then the Table II gives the ratio of π*(n) to π (n). inequality and 푥 > 1 for the upper inequality. Table-II: . Ratio of π*(n) to π (n) Ratio n π(n) π*(n) 102 25 5 0.2 B. On Distribution of Primes 103 168 36 0.21428 It is a fact that till date there are 183 different proofs for the Euclid’s observation that primes are infinitely many. So by 104 1229 261 0.21236 beginning with 푝 = 2, if we presume that 푝 , 푝 , … , 푝 are 1 1 2 푘 From Table II it is clear that π*(n) ∼ α π (n) where α ≈ defined then by selecting a prime 푝 , one can continue the 푘+1 ∗ 푛 푘 0.21. Then by PNT we can say that 휋 (n)~α where α ≈ sequence for which N 푗 =1 푝푗 + 1 ≡ 0 푚표푑 푝푘+1 . Clearly ln 푛 0.21. Thus, we can enhance the approximation for the such a sequence of selection 푝푗 is not unique as there are property of being y to be prime as ln n. The Table III presents many options for 푝푘+1 to keep the above relation intact. Mullin [7] proposed two canonical options namely the least a comparison of the actual number to the estimated number of primes we considered. The Fig.1. represents the plot for 푝푘+1 and the greatest 푝푘+1. The former resulted in a sequence 2,3,7, 43, 13,53,5,6221671,38709183810571, Table III. 139,2081,11,17,5471, … [ 8] called first EM-sequence and Table-III: Comparison of Estimates the latter in 2,3,7,43,139,50207,340999, 2365347734339, … Actual Ratio n Expected [9] called the second EM-sequence. Mullin then asked number number whether these two sequences include in it every prime. In 102 5 4.56 1.096 [10] it was conjectured relaying on stochastic arguments that the second EM sequence would in all possibility could 103 36 30.4006 1.1841 exhaust all primes by having them in their length but till date not much is known about the first EM sequence. Later the 4 second EM sequence was probed further in [11] and the 10 261 228.004 1.1447 authors reported that all of 5,11,13,17,19,23,29,31,37,41 푎푛푑 47 are missing and they conjectured that infinitely many primes do not occur in the second EM sequence. The authors in [12] confirmed this conjecture in positive. Understandably they used quadratic reciprocity for the Jacobi symbol and made a good use of the works of Burgess on short character sums’ upper bound. In this article, we are discussing about the primes of the 푘 form 푦 = 2 푖=1 푥푖 + 1; 푥푖 ∈ 푃 − 2 ; 푥푖 ′푠 are distinct. It is to be noted that if these 푥푖 ′푠 are consecutive primes, then we obtain primorial primes.

Published By: Retrieval Number: L36521081219/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.L3652.1081219 4151 & Sciences Publication International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019

−푔(푥2)푔(푥) 푔 푥2 ≅ 푑푥 . Another estimate of variation in 푥 density comes with the help of derivatives namely, 푔 푥 + 푑푥 2 − 푔 푥2 ≅ 푔′ 푥2 2푥푑푥. So 푔′ 푥2 = −푔(푥2)푔(푥)/ 2푥2 . A change of variable from 푥2 푡표 푥 yields 푔′ 푥 = −푔(푥)푔( 푥) . 2푥 The randomness in prime number distribution can be stochastic or deterministic. As the prime numbers’ distribution behaves Poissonian-like there is better chance that they are stochastically distributed [15]. The authors in [16] however presume that prime numbers may be eigenvalues of a quantum system. A primary reason for attempting to decipher the pattern of distribution of prime Fig.1. Variation of Actual to Expected Number of numbers is that one can compare them with the hidden or Primes of the particular case known patterns in the computational nature of brains, where It is noted that the heuristic estimates for the Primorial and the natural numbers have a significant role. The randomness Factorial primes were determined by Caldwell and Gallot of neuron signals is similar to that of prime numbers. [13]. According to them, both the Primorial and Factorial primes have the same heuristic estimate given by the formula IV. OPEN PROBLEMS eγ ln(N) where N is a fixed real number. We compare these In this section we point out some open problems existing in ratios with the ratio of present study in Table IV. the study of prime numbers in applied areas like control systems and bio-networks. Table-IV: Comparison with Primorial and Factorial primes (i) Prove or Disprove the theoretical nature of the prime Primorial Factorial numbers. n Present Estimate Estimate (ii) Test whether any intermittent designs are there in the [7] [7] 10 0 4.1 4.1 sequence of prime numbers. (iii) Is it possible to relate patterns detected in neuron signals 102 4.56 8.2 8.2 with the prime number pattern to describe the computational device of the brains. 103 30.4006 12.3 12.3 104 228.004 16.4 16.4 (iv) How to describe the source of interspike breaks anomaly to provide for the sequential components of the neuronal code III. APPLICATIONS (v) How to compare the impulsive movement of the hippocampal singular neurons with the randomness of the A system that deploys measurements to self-evaluate its prime numbers. behavior for the purpose of better imminent behavior is called a feedback and control system (FCS). According to [14] FCS V. CONCLUSIONS can be described by differential equations and prime numbers are some type of fcs. This thought process has occurred to In the present study we discussed on prime numbers, their them on seeing the random distribution of prime numbers distribution and some applications. Also, we introduced some 푘 along the integer number line. Much earlier to this numerical analysis on primes of the form = 2 푖=1 푥푖 + observation Gauss remarked that the density of primes is 1; 푥푖∈푃−2; 푥푖′푠 are distinct. We compared our findings with inversely proportional to ln 푥. Using Gauss’s remark, the primorial as well as factorial primes. Then we discussed function 휋(푥) was coined to denote the number of primes not certain interesting application regarding the effectiveness of exceeding 푥. To comprehend FCS mechanism of primes, it primes and their distributions in control system and Bio was first observed that a presence of a large number of primes Engineering. Also, we pointed out some open problems in in one given interval affects that much presence in a engineering fields other than cyber security and subsequent larger interval while its meagre presence in a cryptography. given interval boosts its chance of presence in larger numbers in higher intervals. So, by following Eratosthenes sieve ACKNOWLEDGMENT process one can see early that the removal process of primes The authors gratefully acknowledge Tata Realty-SASTRA alters the density by a factor of 1 − 1/푝. In [14] the authors Srinivasa Ramanujan Research Chair Grant for funding this nicely derived the following differential equation that models research work. the density of primes at 푥. Suppose that 푔(푥) represents a differentiable function REFERENCES that roughly describes density of primes in the nearest neighborhood of x. Then the change in density on an interval 1. A Borning, “Some results for k! + 1 and 2 × 3 × 5 × · · · × P + 1” , 2 2 2 Mathematics of Computation, 1972, pp.567–570. with end points 푥 and (푥 + 푑푥) is 푔 푥 + 푑푥 − 2. M. Templer. “On the Primality of k! + 1 and 2 × 3 × 5 × · · · ×P+ 1”, 2 푔 푥 . Note that every prime in (푥, 푥 + 푑푥) changes the Mathematics of Computation. 1980, vol. 34, pp.303–304. density on (푥2, (푥 + 푑푥)2) by a factor of 1 − 1/푥. So, every prime remove 푔(푥2)/푥 from the density. As there are

푔(푥)푑푥 primes in (푥, 푥 + 푑푥), a first approximation in the 2 2 2 density variation on (푥 , (푥 + 푑푥) ) is 푔 푥 + 푑푥 −

Published By: Retrieval Number: L36521081219/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.L3652.1081219 4152 & Sciences Publication

On Prime Numbers and Related Applications

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AUTHORS PROFILE Veena Narayanan is a research student in SASTRA. She enrolled for Ph.D. in 2017. Her area of research is Number Theory. She completed her schoolings in Kerala. She had secured third rank in Bachelor degree and First rank in Master degree under Calicut University, Kerala. She has one year of teaching experience and two years of research experience. Her research is funded by TATA Realty SASTRA Srinivasa Ramanujan Research chair grant.

V.Yegnanarayanan obtained his Full Time PhD in Mathematics from Annamalai University in Nov 1996. He has 31 years of total experience in which 16 years as Professor & Dean/HOD. He has also worked as a Visiting Scientist on lien in TIFR and IMSC. He has authored 164 research papers and guided 6 students to Ph.D. degree. He has delivered a number of invited talks, organized funded conferences, FDP etc., did a lot of review work for MR, zbMATH and reputed journals, completed research projects funded by NBHM-GOI, won Sentinel of Science Award from Publons. TN State (Periyar) University has recognized his qualifications and experience for the post of Principal in 2009.

R Srikanth is a chair professor for Number Theory under TATA Realty-SASTRA Srinivasa Ramanujan Research chair. His area of specialization is Number Theory and allied areas. He had a lot of experience in teaching as well as in Research for many years both in core subject and applied area. He received his Doctoral degree (Mathematics, Number theory) from Bharathidasan University, Trichy in 2006. He has completed one DST project on “Optimal Rehabilitation and Expansion of Water Distribution Networks” from 2008 to 2010 and a DRDO sanctioned project on “A Novel methodology for the identification of newer elliptic curve that are band suited to cryptosystem” from2013-2015.

Published By: Retrieval Number: L36521081219/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.L3652.1081219 4153 & Sciences Publication